Asymptotic Representations of Quantum Affine Superalgebras

Huafeng Zhang

Symmetry, integrability and geometry: methods and applications, Tome 13 (2017)

Résumé

We study representations of the quantum affine superalgebra associated with a general linear Lie superalgebra. In the spirit of Hernandez–Jimbo, we construct inductive systems of Kirillov–Reshetikhin modules based on a cyclicity result that we established previously on tensor products of these modules, and realize their inductive limits as modules over its Borel subalgebra, the so-called $q$-Yangian. A new generic asymptotic limit of the same inductive systems is proposed, resulting in modules over the full quantum affine superalgebra. We derive generalized Baxter's relations in the sense of Frenkel–Hernandez for representations of the full quantum group.

Détail

Keywords: quantum groups; superalgebras; asymptotic representations; Baxter operators.

Bibliographie

[1] Baxter R. J., “Partition function of the eight-vertex lattice model”, Ann. Physics, 70 (1972), 193–228 | DOI | MR | Zbl

[2] Bazhanov V. V., Hibberd A. N., Khoroshkin S. M., “Integrable structure of ${\mathcal W}_3$ conformal field theory, quantum Boussinesq theory and boundary affine Toda theory”, Nuclear Phys. B, 622 (2002), 475–547, arXiv: hep-th/0105177 | DOI | MR | Zbl

[3] Bazhanov V. V., Lukyanov S. L., “Integrable structure of quantum field theory: classical flat connections versus quantum stationary states”, J. High Energy Phys., 2014:9 (2014), 147, 68 pp., arXiv: 1310.4390 | DOI | MR | Zbl

[4] Bazhanov V. V., Lukyanov S. L., Zamolodchikov A. B., “Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz”, Comm. Math. Phys., 177 (1996), 381–398, arXiv: hep-th/9412229 | DOI | MR | Zbl

[5] Bazhanov V. V., Lukyanov S. L., Zamolodchikov A. B., “Integrable structure of conformal field theory. II. ${\rm Q}$-operator and DDV equation”, Comm. Math. Phys., 190 (1997), 247–278, arXiv: hep-th/9604044 | DOI | MR | Zbl

[6] Bazhanov V. V., Lukyanov S. L., Zamolodchikov A. B., “Integrable structure of conformal field theory. III. The Yang–Baxter relation”, Comm. Math. Phys., 200 (1999), 297–324, arXiv: hep-th/9805008 | DOI | MR | Zbl

[7] Bazhanov V. V., Tsuboi Z., “Baxter's Q-operators for supersymmetric spin chains”, Nuclear Phys. B, 805 (2008), 451–516, arXiv: 0805.4274 | DOI | MR | Zbl

[8] Beisert N., Galleas W., Matsumoto T., “A quantum affine algebra for the deformed Hubbard chain”, J. Phys. A: Math. Theor., 45 (2012), 365206, 20 pp., arXiv: 1102.5700 | DOI | MR | Zbl

[9] Benkart G., Kang S.-J., Kashiwara M., “Crystal bases for the quantum superalgebra $U_q({\mathfrak{gl}}(m,n))$”, J. Amer. Math. Soc., 13 (2000), 295–331, arXiv: math.QA/9810092 | DOI | MR | Zbl

[10] Damiani I., “La $R$-matrice pour les algèbres quantiques de type affine non tordu”, Ann. Sci. École Norm. Sup. (4), 31 (1998), 493–523 | DOI | MR | Zbl

[11] Faddeev L. D., Reshetikhin N. Yu., Takhtajan L. A., “Quantization of Lie groups and Lie algebras”, Yang–Baxter equation in Integrable Systems, Advanced Series in Mathematical Physics, 10, ed. M. Jimbo, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989, 299–309 | DOI | MR

[12] Felder G., “Elliptic quantum groups”, XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 1995, 211–218, arXiv: hep-th/9412207 | MR | Zbl

[13] Felder G., Zhang H., “Baxter operators and asymptotic representations”, Selecta Math. (N.S.) (to appear) , arXiv: 1611.00628 | DOI

[14] Frenkel E., Hernandez D., “Baxter's relations and spectra of quantum integrable models”, Duke Math. J., 164 (2015), 2407–2460, arXiv: 1308.3444 | DOI | MR | Zbl

[15] Frenkel E., Mukhin E., “The Hopf algebra ${\rm Rep}\,U_q\widehat{\mathfrak{gl}}_\infty$”, Selecta Math. (N.S.), 8 (2002), 537–635, arXiv: math.QA/0103126 | DOI | MR | Zbl

[16] Frenkel E., Reshetikhin N., “The $q$-characters of representations of quantum affine algebras and deformations of $\mathcal W$-algebras”, Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999, 163–205, arXiv: math.QA/9810055 | DOI | MR | Zbl

[17] Gautam S., Toledano Laredo V., “Yangians, quantum loop algebras, and abelian difference equations”, J. Amer. Math. Soc., 29 (2016), 775–824, arXiv: 1310.7318 | DOI | MR | Zbl

[18] Heckenberger I., Spill F., Torrielli A., Yamane H., “Drinfeld second realization of the quantum affine superalgebras of $D^{(1)}(2,1;x)$ via the Weyl groupoid”, Combinatorial Representation Theory and Related Topics, RIMS Kôkyûroku Bessatsu, B8, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, 171–216, arXiv: 0705.1071 | MR

[19] Hernandez D., “Representations of quantum affinizations and fusion product”, Transform. Groups, 10 (2005), 163–200, arXiv: math.QA/0312336 | DOI | MR | Zbl

[20] Hernandez D., “Simple tensor products”, Invent. Math., 181 (2010), 649–675, arXiv: 0907.3002 | DOI | MR | Zbl

[21] Hernandez D., Jimbo M., “Asymptotic representations and Drinfeld rational fractions”, Compos. Math., 148 (2012), 1593–1623, arXiv: 1104.1891 | DOI | MR | Zbl

[22] Ip I. C.-H., Zeitlin A. M., “Q-operator and fusion relations for $U_q(C^{(2)}(2))$”, Lett. Math. Phys., 104 (2014), 1019–1043, arXiv: 1312.4063 | DOI | MR | Zbl

[23] Kac V. G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990 | DOI | MR | Zbl

[24] Kulish P. P., Zeitlin A. M., “Superconformal field theory and SUSY $N=1$ KdV hierarchy. II. The $Q$-operator”, Nuclear Phys. B, 709 (2005), 578–591, arXiv: hep-th/0501019 | DOI | MR | Zbl

[25] Mukhin E., Young C. A.S., “Affinization of category $\mathcal{O}$ for quantum groups”, Trans. Amer. Math. Soc., 366 (2014), 4815–4847, arXiv: 1204.2769 | DOI | MR | Zbl

[26] Palev T. D., Stoilova N. I., Van der Jeugt J., “Finite-dimensional representations of the quantum superalgebra $U_q[{\rm gl}(n/m)]$ and related $q$-identities”, Comm. Math. Phys., 166 (1994), 367–378, arXiv: hep-th/9306149 | DOI | MR | Zbl

[27] Perk J. H. H., Schultz C. L., “New families of commuting transfer matrices in $q$-state vertex models”, Phys. Lett. A, 84 (1981), 407–410 | DOI | MR

[28] Reshetikhin N.Yu., Takhtadzhyan L. A., Faddeev L. D., “Quantization of Lie groups and Lie algebras”, Leningrad Math. J., 1 (1990), 193–225 | MR | Zbl

[29] Tan Y., Guay N., “Local Weyl modules and cyclicity of tensor products for Yangians”, J. Algebra, 432 (2015), 228–251, arXiv: 1503.06510 | DOI | MR | Zbl

[30] Tsuboi Z., “Analytic Bethe ansatz and functional equations for Lie superalgebra ${\rm sl}(r+1|s+1)$”, J. Phys. A: Math. Gen., 30 (1997), 7975–7991, arXiv: 0911.5386 | DOI | MR | Zbl

[31] Tsuboi Z., “Asymptotic representations and $q$-oscillator solutions of the graded Yang–Baxter equation related to Baxter Q-operators”, Nuclear Phys. B, 886 (2014), 1–30, arXiv: 1205.1471 | DOI | MR | Zbl

[32] Yamane H., “Quantized enveloping algebras associated with simple Lie superalgebras and their universal $R$-matrices”, Publ. Res. Inst. Math. Sci., 30 (1994), 15–87 | DOI | MR | Zbl

[33] Zhang H., “Representations of quantum affine superalgebras”, Math. Z., 278 (2014), 663–703, arXiv: 1309.5250 | DOI | MR | Zbl

[34] Zhang H., “RTT realization of quantum affine superalgebras and tensor products”, Int. Math. Res. Not., 2016 (2016), 1126–1157, arXiv: 1407.7001 | DOI | MR | Zbl

[35] Zhang H., “Fundamental representations of quantum affine superalgebras and $R$-matrices”, Transform. Groups, 22 (2017), 559–590, arXiv: 1506.06093 | DOI | MR

[36] Zhang H., Length two representations of quantum affine superalgebras and Baxter operators, arXiv: 1612.09476

[37] Zhang H., Elliptic quantum groups and Baxter relations, arXiv: 1706.07574